By Michel Brion (auth.), H. E. A. Campbell, Aloysius G. Helminck, Hanspeter Kraft, David Wehlau (eds.)

A workforce of Gerry Schwarz’s colleagues and collaborators accrued on the Fields Institute in Toronto for a mathematical festschrift in honor of his sixtieth birthday. This quantity is an outgrowth of that occasion, protecting the wide variety of arithmetic to which Gerry Schwarz has both made primary contributions or prompted others to pursue. The articles are a sampling of contemporary day algebraic geometry with linked staff activities from its major specialists, with a specific specialise in attribute zero and modular invariant theory.

Contributors:

M. Brion

A. Broer

D. Daigle

J. Elmer

P. Fleischmann

G. Freudenberg

D. Greb

P. Heinzner

A. Helminck

B. Kostant

H. Kraft

R. J. Shank

W. Traves

N. R. Wallach

D. Wehlau

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Amer. Math. Soc. 337 (1993), 211–218. On Chevalley–Shephard–Todd’s Theorem in Positive Characteristic Abraham Broer Summary. Let G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the direct summand property holds if there is a surjective k[V ]G -linear map π : k[V ] → k[V ]G . The following Chevalley–Shephard–Todd type theorem is proved.

We assume this; so m = pm for some integer m and we define V to be the quotient module V˜ / v with dimension n := m − 2 ≥ 3. p e p( j−1)+i , then v = ∑mj=1 v j and each v j ∈ V˜ . For 1 ≤ j ≤ m define v j := ∑i=1 ˜ ˜ Write U1 = v1 , . . , vm ⊂ V with image U1 in V . 32 A. Broer We remark that if for σ ∈ Sm and i it holds that σ (vi ) = vi , then since m ≥ 5 we have σ (vi ) − vi ∈ v . ,m p} S p × S p × · · ·× S p . ˜ Suppose p odd or p = 2 and m is even. Then w := ∑m i=1 iei ∈ V and we define U˜ = U˜1 + w with image U ⊂ V .

Then there are two nonsingular vectors u1 , u2 such that the point-stabilizer in + ⊥ GO− n (q) of u1 , u2 is GOn−2 (q) acting irreducibly on u1 , u2 . And we can reduce to that case. (IV) (Symmetric groups) Sn+2 , p|(n + 2), n ≥ 3. Let W = km be a vector space over a field of characteristic p > 0 with basis e1 , . . , em ; we assume m ≥ 5. The symmetric group Sm acts on W by permuting the basis elements. The submodule of codimension one V˜ = ei − e j ; 1 ≤ i < j ≤ m contains the submodule spanned by v = ∑m i=1 ei if and only if p divides m.